\newproblem{lay:7_1_29}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 7.1.29}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Suppose $A$ is invertible and orthogonally diagonalizable. Explain why $A^{-1}$ is also orthogonally diagonalizable.
}{
   % Solution
	If $A$ is orthogonally diagonalizable, then $A=PDP^T$. Then,
	\begin{center}
		$A^{-1}=(PDP^T)^{-1}=(P^T)^{-1}D^{-1}P^{-1}=PD^{-1}P^T$
	\end{center}
	So, $A^{-1}$ is orthogonally diagonalizable.
}
\useproblem{lay:7_1_29}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

